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# There are 2 available positions and 50 candidates, one half of whom ar

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Math Expert
Joined: 02 Sep 2009
Posts: 58429
There are 2 available positions and 50 candidates, one half of whom ar  [#permalink]

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24 Jul 2015, 00:47
00:00

Difficulty:

35% (medium)

Question Stats:

70% (01:36) correct 30% (01:23) wrong based on 216 sessions

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There are 2 available positions and 50 candidates, one half of whom are democrats and another half are republicans. If it was decided that the positions would be filled at random, then what is the probability that the both positions will be taken by members of just one party?

A. 1/25
B. 12/49
C. 1/4
D. 24/49
E. 1/2

Kudos for a correct solution.

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There are 2 available positions and 50 candidates, one half of whom ar  [#permalink]

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Updated on: 26 Jul 2015, 20:34
1
1
Bunuel wrote:
There are 2 available positions and 50 candidates, one half of whom are democrats and another half are republicans. If it was decided that the positions would be filled at random, then what is the probability that the both positions will be taken by members of just one party?

A. 1/25
B. 12/49
C. 1/4
D. 24/49
E. 1/2

Kudos for a correct solution.

Solution -

There are 25 democrats and 25 republicans.

The probability that the both positions will be taken by democrats or republicans = (25C2 + 25C2)/50C2 = 600/1225 = 24/49. ANS D.
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Thanks,

Originally posted by balamoon on 24 Jul 2015, 03:01.
Last edited by balamoon on 26 Jul 2015, 20:34, edited 1 time in total.
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Re: There are 2 available positions and 50 candidates, one half of whom ar  [#permalink]

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24 Jul 2015, 03:36
Bunuel wrote:
There are 2 available positions and 50 candidates, one half of whom are democrats and another half are republicans. If it was decided that the positions would be filled at random, then what is the probability that the both positions will be taken by members of just one party?

A. 1/25
B. 12/49
C. 1/4
D. 24/49
E. 1/2

Kudos for a correct solution.

Let R stand for republicans and D for democrats

Thus the possible combinations are RR or DD

Thus the probability = 2(25/50 * 24/49) = 24/49. D is the correct answer.
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Re: There are 2 available positions and 50 candidates, one half of whom ar  [#permalink]

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24 Jul 2015, 03:39
balamoon wrote:
Bunuel wrote:
There are 2 available positions and 50 candidates, one half of whom are democrats and another half are republicans. If it was decided that the positions would be filled at random, then what is the probability that the both positions will be taken by members of just one party?

A. 1/25
B. 12/49
C. 1/4
D. 24/49
E. 1/2

Kudos for a correct solution.

Solution -

There are 25 democrats and 25 republicans.

The probability that the both positions will be taken by democrats or republicans = (25C2 +25C2)/50C2 = 300/1225 = 12/49. ANS B.

The text marked in red above is not calculated correctly. It should be 24/49 and not 12/49.

(25C2 + 25C2)/50C2 = 600/1225 = 24/49
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Re: There are 2 available positions and 50 candidates, one half of whom ar  [#permalink]

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24 Jul 2015, 05:22
1
Probability that both positions will be taken by Democrats = 25C2/50C2 = 600/2450 = 12/49.
Probability that both positions will be taken by Republicans = 25C2/50C2 = 600/2450 = 12/49.

Required probability = 12+12/49 = 24/49. Ans (D).
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Re: There are 2 available positions and 50 candidates, one half of whom ar  [#permalink]

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24 Jul 2015, 09:00
Bunuel wrote:
There are 2 available positions and 50 candidates, one half of whom are democrats and another half are republicans. If it was decided that the positions would be filled at random, then what is the probability that the both positions will be taken by members of just one party?

A. 1/25
B. 12/49
C. 1/4
D. 24/49
E. 1/2

Kudos for a correct solution.

(25C2 + 25C2) / 50C2 = 24/49. Answer D

But I always wonder.. how can we find out which way to choose: Combinatorics as I just did or:

(2/25 * 1/24) * 2 ... why will this one give us an incorrect result.?. it is quite hard for me to understand ...
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There are 2 available positions and 50 candidates, one half of whom ar  [#permalink]

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24 Jul 2015, 10:22
1
alternative solution that doesnt involve large numbers:

probability of one party having both spots:

(1/2) * (24/49) = 12/49

(1/2) or (25/50) because it does not matter which party or which person gets the first spot. (24/49) because after one person from a particular party is chosen, there are 24 members of the same party left out of 49 total candidates.

Since this result can happen for both parties, (12/49) + (12/49) = (24/49)

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There are 2 available positions and 50 candidates, one half of whom ar  [#permalink]

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24 Jul 2015, 12:08
1
Bunuel wrote:
There are 2 available positions and 50 candidates, one half of whom are democrats and another half are republicans. If it was decided that the positions would be filled at random, then what is the probability that the both positions will be taken by members of just one party?

A. 1/25
B. 12/49
C. 1/4
D. 24/49
E. 1/2

Kudos for a correct solution.

We can select 2 members of one particular type:
25/1 * 24/2 * 2

We can select 2 members of any type:
50/1 * 49/2

P(both positions will be taken by members of just one party) = 25*24*2 / 50*49 = 24/49

Ans. D. 24/49
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Re: There are 2 available positions and 50 candidates, one half of whom ar  [#permalink]

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24 Jul 2015, 12:14
LaxAvenger wrote:
Bunuel wrote:
There are 2 available positions and 50 candidates, one half of whom are democrats and another half are republicans. If it was decided that the positions would be filled at random, then what is the probability that the both positions will be taken by members of just one party?

A. 1/25
B. 12/49
C. 1/4
D. 24/49
E. 1/2

Kudos for a correct solution.

(25C2 + 25C2) / 50C2 = 24/49. Answer D

But I always wonder.. how can we find out which way to choose: Combinatorics as I just did or:

(2/25 * 1/24) * 2 ... why will this one give us an incorrect result.?. it is quite hard for me to understand ...

The text in red is incorrect. How are you able to write the total combinations/ways as : 1/25 or even 1/24 (how can the total ways be fractions?)? Can you show some steps by which you got these numbers?
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Re: There are 2 available positions and 50 candidates, one half of whom ar  [#permalink]

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24 Jul 2015, 17:18
1
Hi All,

In these types of questions, there are usually several different ways to do the 'math' involved. It's important to be on the lookout for any conceptual 'shortcuts' that are built into the question though - the GMAT rarely requires that you perform a really complex calculation to get to the solution.

Here, we're told that there are 25 Democrats and 25 Republicans. We're asked for the probability that two elected positions will go to the SAME party.

Since the calculation 'hinges' on the party of the second person elected matching the party of the first person elected, the first person COULD come from EITHER party (in math terms, the first person elected does NOT matter - the issue is "does the second person come from the same party as the first person?").

Once 1 Democrat OR 1 Republican is elected, there will be only 24 members of that party remaining (and 49 total people remaining. Thus, the probability that the second person comes from the same party as the first is...

24/49

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Rich
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Re: There are 2 available positions and 50 candidates, one half of whom ar  [#permalink]

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25 Jul 2015, 03:35
Bunuel wrote:
There are 2 available positions and 50 candidates, one half of whom are democrats and another half are republicans. If it was decided that the positions would be filled at random, then what is the probability that the both positions will be taken by members of just one party?

A. 1/25
B. 12/49
C. 1/4
D. 24/49
E. 1/2

Kudos for a correct solution.

Both candidates will either be Democrats or Repub.

Required probability= (25C2+25C2)/50C2=600/25*49=24/49
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Posts: 58429
Re: There are 2 available positions and 50 candidates, one half of whom ar  [#permalink]

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26 Jul 2015, 12:26
Bunuel wrote:
There are 2 available positions and 50 candidates, one half of whom are democrats and another half are republicans. If it was decided that the positions would be filled at random, then what is the probability that the both positions will be taken by members of just one party?

A. 1/25
B. 12/49
C. 1/4
D. 24/49
E. 1/2

Kudos for a correct solution.

800score Official Solution:

Let’s consider an outcome to be an ordered pair (x, y), where x is a member, who took position #1, y is a member, who took position #2. There are 50 × 49 possible outcomes.

There are 25 × 24 favorable outcomes for one party (x and y are from the same party). So there are 2 × (25 × 24) favorable outcomes in total.

The probability is (2 × 25 × 24) / (50 × 49) = 24/49.

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Re: There are 2 available positions and 50 candidates, one half of whom ar  [#permalink]

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Re: There are 2 available positions and 50 candidates, one half of whom ar   [#permalink] 13 Oct 2018, 04:00
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